Solid State Theory. (Theorie der kondensierten Materie) Wintersemester 2015/16


 Dwain Butler
 1 years ago
 Views:
Transcription
1 Martin Eckstein Solid State Theory (Theorie der kondensierten Materie) Wintersemester 2015/16 MaxPlanck Research Departement for Structural Dynamics, CFEL Desy, Bldg. 99, Rm Luruper Chaussee Hamburg, Germany Tel: +49 (0) Lecture Website: Literature: G. Czychol, Theoretische Festkörperphysik, Springer J. M. Ziman, Principles of the Theory of Solids, Cambridge University Press, N. W. Ashcroft and M. D. Mermin, Solid State Physics. A. Altland and B. Simons, Condensed Matter Field Theory, Cambridge University Press, Topics: Electrons in the periodic crystal (bandstructure calculations); Interacting electrons: Screening, Landau s Theory of Fermi Liquids; Phonons; Superconductivity; Magnetism; 1
2 2
3 Introduction In solid state physics, we aim to understand structure of solids phases of condensed matter systems, like metallic or insulating behavior, magnetism, superconductivity,... elementary excitations ( spectroscopical measurements, transport properties) For the understanding of many phenomena, quantum mechanics is essential. Examples: chemical binding and stability of matter explained by quantum effect. Our nonrelativistic theory of everything would be given by the Hamiltonian H = p 2 i 2m + e 2 r i i<j i r j electrons + i P 2 i e 2 Z i Z j R i R j 2m + i<j nuclei e 2 Z k r i R k i,k en attraction (1) The typical binding energy of en electron to a nucleus is of the order of several electron volts (Hatom: 13.6eV =1Ry). With the basic energytemperature conversion 1eV = k B Kelvin 300K 30meV 1Ry, (2) one can see that thermal fluctuations k B T are negligible compared to atomic energies at room temperature. The Hamiltonian (1) is thus useful for the understanding of properties of matter at K (this would be a plasma), but it is usually not a good starting point to learn about a solid at room temperature. Not even fundamental phenomena such as the formation of a crystal can be predicted fully abinitio, although it should be contained in the formulation. The Hamiltonian can be vied as the highenergy theory for condensed matter physics, from which lowenergy effective theories for condensed matter phases emerge. The situation in low energy physics is opposite to highenergy physics, where one tries to uncover the highenergy theories from which the physical laws for atoms and nuclei emerge: 3
4 10 mev condensed matter: effective theories 10 ev atoms and nuclei MeV highenergy physics new ground states with new types of excitations (quasiparticles) and effective interactions: this lecture Examples for effective theories for the solid: Harmonic crystal Spin models ( magnetism of localized moments) Electronic quasiparticles 4
5 Periodic structures Most solids have crystalline order on the atomic scale (although, macroscopically they can form crystallites on the µm scale.) Exceptions are glasses (amorphous structures), liquid crystals, etc., see Fig. 1. Crystalline symmetries are essential for a theoretical description of solids, hence we will deal almost exclusively with crystalline structures in this lecture. Figure 1: Left: Translational, but no orientational symmetry (e.g., Spin glass). Middle: Orientational, but no translational symmetry (e.g., nematic order). Right: Perfect crystalline order. Classification of crystalline structures Bravais lattice: Infinite set of points generated via translations of one point by integer linear combinations of a set of given vectors a i { R} = {n 1 a 1 + n 2 a 2 + n 3 a 3 n i Z}, (3) a i primitive lattice vectors, linearly independent. Lattice looks identical from every of its points. Example: The vectors a 0 0 a 1 = 0,a 2 = a,a 3 = a generate the perfect cubic lattice with lattice spacing a. 5
6 Note: The choice of the primitive lattice vectors is not unique: If a i is a set of primitive vectors, a i generates the same lattice iff a i = j M ija i with an integer matrix M of determinant ±1. Primitive unit cell: The volume spanned by the primitive unit vectors is called the primitive unit cell. It has volume V = a 1 (a 2 a 3 ). (4) In general, a unit cell can be any volume that covers full space without overlaps when it is translated by every vector of the Bravais lattice. A special (uniquely defined) choice is the Wigner Seitz cell, which is chosen to be the set of all points closer to a point R of the Bravais lattice than any other point (see Fig. 2) Figure 2: Left: A primitive unit cell of the square lattice (which has not the full point group symmetry of the lattice) Right: The WignerSeitz cell. The WignerSeitz cell around a lattice point R is geometrically constructed by drawing the line from R to any other point of the lattice, and the plane perpendicular to the line halfway between the points. All these planes cut out the Wigner Seitz cell. Crystal Structure: A Bravais lattice is only the set of points generated by the translation operations. The full crystal structure is obtained by placing objects of typically lower symmetry (atoms, molecules) in the lattice. The objects within one unit cell is called the basis. Example: The honeycomb lattice is not a Bravais lattice, but a Bravais lattice with a basis of two atoms (Fig. 3) Figure 3: The honeycomb lattice. 6
7 Crystal Symmetries: A crustal symmetry is a linear transformation of space r r = Dr + a (5) which maps the structure onto itself. a : Translation. D : orthogonal transformation (rotation, reflection, inversion). All operations D define the point group of the crystal (D is a transformation which leaves one point of the lattice invariant). All symmetry operations (point group + translations) define the space group of the crystal. Note: The operations D do not necessarily form a subgroup of the space group; there can be rotations/reflections that are symmetries of the crystal only in combination with a translation: screw axis: rotation + translation along rotation axis glide plane: reflection + translation along reflection plane a 2 a 1 dashed line: glide plane (reflection + translation by a1/2) How many crystal symmetries are there? Important restriction: only 2,3,4,6 fold rotation axis is compatible with translational symmetry (Algebraic) proof: Consider rotation matrix around z: cos(ϕ) sin(ϕ) 0 D z (ϕ) = 0sin(ϕ) cos(ϕ) Trace of this matrix is invariant under transformation into a different basis, because Tr[U 1 DU] = TrD for any invertible matrix U. Transforming to the basis defined by the primitive vectors (note 7
8 that the matrix U above need not be orthogononal), the rotation must be a matrix with integer entries (it maps primitve vectors on primitive vectors). Hence D z must satisfy Tr D z (ϕ) =2cos(ϕ)+1! Z. The only solutions to this equation are ϕ =0, π 3, π 2, 2π 3,π. Note: There are structureswith apparent 5fold coordination in reziprocal space (Braggpeaks in Xray diffraction, see below). These quasicrystals are not periodic in space, but may be obtained by projecting a periodic structure in higher dimensional space onto a subspace along a direction not commensurate with the primitive vectors. The crystrallographic restriction finally restricts the number of Bravais lattices and crystal structures (i.e., Bravais lattice + basis): Bravais lattice crystal structure number of point groups 7 (4) 32 (13) number of space groups 14 (5) 230 (17) The number in brackets corresponds to spacial dimension d =2. The space groups in d =2are called wallpaper groups. The correspond to a symmetries of mosaic tilings of the plane. There are many good illustrations in the internet, e.g., https://de.wikipedia.org/wiki/ebene kristallographische Gruppe For a more detailed discussion of symmetries and Bravais lattices in 3d, see Ashcroft & Mermin, Chapter 6. Importance of symmetry for the description of solids: Quantum numbers, degeneracies in the spectrum (see band structure lectures). Symmetry determines the response coefficients of a solid. In general, the physical observables must be symmetric under all crystal symmetries. Example: In general, the conductivity σ is given by the linear response relation j α = α σ αα E α, α,α = x, y, z. 8 (j and E are current and electric field, respectively). The matrix σ must be invariant under the pointgroup operations of the crystal, DσD 1! = σ.
9 For cubic symmetry, this implies already that the matrix σ is a scalar (proportional to the 1 identity matrix): Rotations by 180 around the z axis, with D = 1, imply 1 σ 11 σ 12 σ 13 σ 11 σ 12 σ 13 σ 11 σ 12 σ 13 D σ 21 σ 22 σ 23 D 1 = σ 21 σ 22 σ 23 =! σ 21 σ 22 σ 23, σ 31 σ 32 σ 33 σ 31 σ 32 σ 33 σ 31 σ 32 σ 33 hence σ 12 = σ 13 = σ 21 = σ 31 =0. Similar rotations around the y and z axis require the matrix to be diagonal. Rotations by 120 around the bodydiagonal (which permute x,y,z) imply that σ 11 = σ 22 = σ 33. The reziprocal lattice Definition: For a Bravais lattice G = { R}, the reziprocal lattice is defined by the points G = { G G R =2πn with n Z for all R G}. (6) The reziprocal lattice is a Bravais lattice itself, with primitive vectors a 2 a 3 b1 =2π a 1 (a 2 a 3 ), a 3 a 1 b2 =2π a 1 (a 2 a 3 ), a 1 a 2 b3 =2π a 1 (a 2 a 3 ). Note: The reziprocal lattice need not be of the same lattice type (fcc bcc). Example: If a i are orthogonal, a 1 = a x ê x, a 2 = a y ê y, a 3 = a z ê z, then b 1 = 2π a x ê x, b 2 = 2π a y ê y, b 3 = 2π a z ê z. a 2 lattice reziprocal lattice a 1 b 2 = 2π a 2 b 1 = 2π a 1 Figure 4: Reziprocal lattice of an orthogonal lattice in dimensions d =2. The reziprocal lattice is important in three aspects, which are described below: 9
10 1) Fouriertransformation of space periodic functions If f(r ) is a function which has the lattice periodicity, i.e., f(r + R)=f(r ) for all R G, then the only nonzero Fourier coefficients of f appear at the points of the reziprocal lattice. In short, f k = d 3 rf(r )e ikr = d 3 rf(r + R)e ikr = d 3 rf(r )e i k(r R) = e i k R f k, (where in the first equality we used translational invariance of f, the second equality is a variable transform), i.e., k must satisfy e i kr =1for! all R G, which is equivalent to k G by the definition (6). In general, the fourier series representation of f and its inverse is given by (V is the volume of the unit cell) f(r )= f G e i Gr G G f G = 1 d 3 rf(r )e i Gr V unitcell (7) (8) 10
11
12
13
14
Crystal Structure. A(r) = A(r + T), (1)
Crystal Structure In general, by solid we mean an equilibrium state with broken translation symmetry. That is a state for which there exist observables say, densities of particles with spatially dependent
More informationSolid State Device Fundamentals
Solid State Device Fundamentals ENS 345 Lecture Course by Alexander M. Zaitsev alexander.zaitsev@csi.cuny.edu Tel: 718 982 2812 Office 4N101b 1 Solids Three types of solids, classified according to atomic
More informationCHAPTER 12 MOLECULAR SYMMETRY
CHAPTER 12 MOLECULAR SYMMETRY In many cases, the symmetry of a molecule provides a great deal of information about its quantum states, even without a detailed solution of the Schrödinger equation. A geometrical
More informationCHAPTER 3: CRYSTAL STRUCTURES
CHAPTER 3: CRYSTAL STRUCTURES Crystal Structure: Basic Definitions  lecture Calculation of material density selfprep. Crystal Systems lecture + selfprep. Introduction to Crystallography lecture + selfprep.
More informationChapter 3: The Structure of Crystalline Solids
Sapphire: cryst. Al 2 O 3 Insulin : The Structure of Crystalline Solids Crystal: a solid composed of atoms, ions, or molecules arranged in a pattern that is repeated in three dimensions A material in which
More informationElementary Crystallography for XRay Diffraction
Introduction Crystallography originated as the science of the study of crystal forms. With the advent of the xray diffraction, the science has become primarily concerned with the study of atomic arrangements
More informationThe quantum mechanics of particles in a periodic potential: Bloch s theorem
Handout 2 The quantum mechanics of particles in a periodic potential: Bloch s theorem 2.1 Introduction and health warning We are going to set up the formalism for dealing with a periodic potential; this
More informationLECTURE NOTES ON THE MATHEMATICS BEHIND ESCHER S PRINTS: FORM SYMMETRY TO GROUPS
LECTURE NOTES ON THE MATHEMATICS BEHIND ESCHER S PRINTS: FORM SYMMETRY TO GROUPS JULIA PEVTSOVA 1. Introduction to symmetries of the plane 1.1. Rigid motions of the plane. Examples of four plane symmetries
More informationCrystals are solids in which the atoms are regularly arranged with respect to one another.
Crystalline structures. Basic concepts Crystals are solids in which the atoms are regularly arranged with respect to one another. This regularity of arrangement can be described in terms of symmetry elements.
More informationThe nearlyfree electron model
Handout 3 The nearlyfree electron model 3.1 Introduction Having derived Bloch s theorem we are now at a stage where we can start introducing the concept of bandstructure. When someone refers to the bandstructure
More informationReview: Vector space
Math 2F Linear Algebra Lecture 13 1 Basis and dimensions Slide 1 Review: Subspace of a vector space. (Sec. 4.1) Linear combinations, l.d., l.i. vectors. (Sec. 4.3) Dimension and Base of a vector space.
More information2.2 Symmetry of singlewalled carbon nanotubes
2.2 Symmetry of singlewalled carbon nanotubes 13 Figure 2.5: Scanning tunneling microscopy images of an isolated semiconducting (a) and metallic (b) singlewalled carbon nanotube on a gold substrate. The
More informationLecture 2. Surface Structure
Lecture 2 Surface Structure Quantitative Description of Surface Structure clean metal surfaces adsorbated covered and reconstructed surfaces electronic and geometrical structure References: 1) Zangwill,
More information5.3 ORTHOGONAL TRANSFORMATIONS AND ORTHOGONAL MATRICES
5.3 ORTHOGONAL TRANSFORMATIONS AND ORTHOGONAL MATRICES Definition 5.3. Orthogonal transformations and orthogonal matrices A linear transformation T from R n to R n is called orthogonal if it preserves
More informationChapter 2 Crystal Lattices and Reciprocal Lattices
Chapter 2 Crystal Lattices and Reciprocal Lattices Abstract In this chapter, the basic unit vectors in real space and the basic unit vectors in reciprocal space, as well as their reciprocal relationships,
More information in a typical metal each atom contributes one electron to the delocalized electron gas describing the conduction electrons
Free Electrons in a Metal  in a typical metal each atom contributes one electron to the delocalized electron gas describing the conduction electrons  if these electrons would behave like an ideal gas
More informationPrinciple of Formation of Magnetic Field of Iron
Principle of Formation of Magnetic Field of Iron Anatoli Bedritsky Abstract. This article discovers physical phenomena which occur at the formation of the magnetic field of iron and the reason of an attraction
More informationCrystal Structures and Symmetry
PHYS 624: Crystal Structures and Symmetry 1 Crystal Structures and Symmetry Introduction to Solid State Physics http://www.physics.udel.edu/ bnikolic/teaching/phys624/phys624.html PHYS 624: Crystal Structures
More information1 Introduction to Matrices
1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns
More informationLinear Algebra Concepts
University of Central Florida School of Electrical Engineering Computer Science EGN3420  Engineering Analysis. Fall 2009  dcm Linear Algebra Concepts Vector Spaces To define the concept of a vector
More informationEarth and Planetary Materials
Earth and Planetary Materials Spring 2013 Lecture 11 2013.02.13 Midterm exam 2/25 (Monday) Office hours: 2/18 (M) 1011am 2/20 (W) 1011am 2/21 (Th) 11am1pm No office hour 2/25 1 Point symmetry Symmetry
More information13 Solutions for Section 6
13 Solutions for Section 6 Exercise 6.2 Draw up the group table for S 3. List, giving each as a product of disjoint cycles, all the permutations in S 4. Determine the order of each element of S 4. Solution
More informationWallpaper Patterns. The isometries of E 2 are all of one of the following forms: 1
Wallpaper Patterns Symmetry. We define the symmetry of objects and patterns in terms of isometries. Since this project is about wallpaper patterns, we will only consider isometries in E 2, and not in higher
More informationPart 5 Space groups. 5.1 Glide planes. 5.2 Screw axes. 5.3 The 230 space groups. 5.4 Properties of space groups. 5.5 Space group and crystal structure
Part 5 Space groups 5.1 Glide planes 5.2 Screw axes 5.3 The 230 space groups 5.4 Properties of space groups 5.5 Space group and crystal structure Glide planes and screw axes 32 point groups are symmetry
More informationPrecession photograph
Precession photograph Spacing of spots is used to get unit cell dimensions. Note symmetrical pattern. Crystal symmetry leads to diffraction pattern symmetry. Groups Total For proteins Laue group 230 space
More informationIntroduction to symmetry
130320  Some handout slides are hidden @13, @43, 3 more at the end. Introduction to symmetry 1 Recap 1/ to understand function, we need structure; to get structure, we need xray crystallography 2/ remarkable
More informationChapter 2: Crystal Structures and Symmetry
Chapter 2: Crystal Structures and Symmetry Laue, ravais December 28, 2001 Contents 1 Lattice Types and Symmetry 3 1.1 TwoDimensional Lattices................. 3 1.2 ThreeDimensional Lattices................
More informationIntroduction to Matrix Algebra
Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra  1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary
More informationSummary of week 8 (Lectures 22, 23 and 24)
WEEK 8 Summary of week 8 (Lectures 22, 23 and 24) This week we completed our discussion of Chapter 5 of [VST] Recall that if V and W are inner product spaces then a linear map T : V W is called an isometry
More informationBasic Concepts of Crystallography
Basic Concepts of Crystallography Language of Crystallography: Real Space Combination of local (point) symmetry elements, which include angular rotation, centersymmetric inversion, and reflection in mirror
More informationLinear Algebra Review. Vectors
Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka kosecka@cs.gmu.edu http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa Cogsci 8F Linear Algebra review UCSD Vectors The length
More informationPhysics 551: Solid State Physics F. J. Himpsel
Physics 551: Solid State Physics F. J. Himpsel Background Most of the objects around us are in the solid state. Today s technology relies heavily on new materials, electronics is predominantly solid state.
More informationSymmetry. From: Lipson/Cochran, The Determination of Crystal Structures
Symmetry A body is said to be symmetrical when it can be divided into parts that are related to each other in certain ways. The operation of transferring one part to the position of a symmetrically related
More informationState of Stress at Point
State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,
More informationSymmetry and Molecular Structures
Symmetry and Molecular Structures Some Readings Chemical Application of Group Theory F. A. Cotton Symmetry through the Eyes of a Chemist I. Hargittai and M. Hargittai The Most Beautiful Molecule  an Adventure
More informationphys4.17 Page 1  under normal conditions (pressure, temperature) graphite is the stable phase of crystalline carbon
Covalent Crystals  covalent bonding by shared electrons in common orbitals (as in molecules)  covalent bonds lead to the strongest bound crystals, e.g. diamond in the tetrahedral structure determined
More informationCHAPTER 3 THE STRUCTURE OF CRYSTALLINE SOLIDS PROBLEM SOLUTIONS
CHAPTER THE STRUCTURE OF CRYSTALLINE SOLIDS PROBLEM SOLUTIONS Fundamental Concepts.6 Show that the atomic packing factor for HCP is 0.74. The APF is just the total sphere volumeunit cell volume ratio.
More informationCRYSTALLINE SOLIDS IN 3D
CRYSTALLINE SOLIDS IN 3D Andrew Baczewski PHY 491, October 7th, 2011 OVERVIEW First  are there any questions from the previous lecture? Today, we will answer the following questions: Why should we care
More informationMATH 304 Linear Algebra Lecture 24: Scalar product.
MATH 304 Linear Algebra Lecture 24: Scalar product. Vectors: geometric approach B A B A A vector is represented by a directed segment. Directed segment is drawn as an arrow. Different arrows represent
More informationIntroduction and Symmetry Operations
Page 1 of 9 EENS 2110 Tulane University Mineralogy Prof. Stephen A. Nelson Introduction and Symmetry Operations This page last updated on 27Aug2013 Mineralogy Definition of a Mineral A mineral is a naturally
More informationAn introduction to the electric conduction in crystals
An introduction to the electric conduction in crystals Andreas Wacker, Matematisk Fysik, Lunds Universitet Andreas.Wacker@fysik.lu.se November 23, 2010 1 Velocity of band electrons The electronic states
More informationCHARACTERISTIC ROOTS AND VECTORS
CHARACTERISTIC ROOTS AND VECTORS 1 DEFINITION OF CHARACTERISTIC ROOTS AND VECTORS 11 Statement of the characteristic root problem Find values of a scalar λ for which there exist vectors x 0 satisfying
More informationUsing determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible:
Cramer s Rule and the Adjugate Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible: Theorem [Cramer s Rule] If A is an invertible
More informationLecture 34: Principal Axes of Inertia
Lecture 34: Principal Axes of Inertia We ve spent the last few lectures deriving the general expressions for L and T rot in terms of the inertia tensor Both expressions would be a great deal simpler if
More informationSpectroscopic Notation
Spectroscopic Notation Most of the information we have about the universe comes to us via emission lines. However, before we can begin to understand how emission lines are formed and what they tell us,
More informationTHE STRUCTURE OF CRYSTALLINE SOLIDS. IE114 Materials Science and General Chemistry Lecture3
THE STRUCTURE OF CRYSTALLINE SOLIDS IE114 Materials Science and General Chemistry Lecture3 Outline Crystalline and Noncrystalline Materials 1) Single, Polycrystalline, Noncrystalline solids 2) Polycrystalline
More informationXray diffraction: theory and applications to materials science and engineering. Luca Lutterotti
Xray diffraction: theory and applications to materials science and engineering Luca Lutterotti luca.lutterotti@unitn.it Program Part 1, theory and methodologies: General principles of crystallography
More informationReciprocal Space and Brillouin Zones in Two and Three Dimensions As briefly stated at the end of the first section, Bloch s theorem has the following
Reciprocal Space and Brillouin Zones in Two and Three Dimensions As briefly stated at the end of the first section, Bloch s theorem has the following form in two and three dimensions: k (r + R) = e 2 ik
More informationby the matrix A results in a vector which is a reflection of the given
Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the yaxis We observe that
More informationA note on the symmetry properties of Löwdin s orthogonalization schemes. Abstract
A note on the symmetry properties of Löwdin s orthogonalization schemes T. A. Rokob Chemical Research Center, ungarian Academy of Sciences, 155 Budapest, POB 17, ungary Á. Szabados and P. R. Surján Eötvös
More informationFor the case of an Ndimensional spinor the vector α is associated to the onedimensional . N
1 CHAPTER 1 Review of basic Quantum Mechanics concepts Introduction. Hermitian operators. Physical meaning of the eigenvectors and eigenvalues of Hermitian operators. Representations and their use. onhermitian
More informationTheory of XRay Diffraction. Kingshuk Majumdar
Theory of XRay Diffraction Kingshuk Majumdar Contents Introduction to XRays Crystal Structures: Introduction to Lattices Different types of lattices Reciprocal Lattice Index Planes XRay Diffraction:
More information6. ISOMETRIES isometry central isometry translation Theorem 1: Proof:
6. ISOMETRIES 6.1. Isometries Fundamental to the theory of symmetry are the concepts of distance and angle. So we work within R n, considered as an innerproduct space. This is the usual n dimensional
More informationRecall that two vectors in are perpendicular or orthogonal provided that their dot
Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal
More informationISOMETRIES OF R n KEITH CONRAD
ISOMETRIES OF R n KEITH CONRAD 1. Introduction An isometry of R n is a function h: R n R n that preserves the distance between vectors: h(v) h(w) = v w for all v and w in R n, where (x 1,..., x n ) = x
More informationMatter, Materials, Crystal Structure and Bonding. Chris J. Pickard
Matter, Materials, Crystal Structure and Bonding Chris J. Pickard Why should a theorist care? Where the atoms are determines what they do Where the atoms can be determines what we can do Overview of Structure
More informationH = = + H (2) And thus these elements are zero. Now we can try to do the same for time reversal. Remember the
1 INTRODUCTION 1 1. Introduction In the discussion of random matrix theory and information theory, we basically explained the statistical aspect of ensembles of random matrices, The minimal information
More informationCondensed Matter Physics Prof. G. Rangarajan Department of Physics Indian Institute of Technology, Madras. Lecture  36 Semiconductors
Condensed Matter Physics Prof. G. Rangarajan Department of Physics Indian Institute of Technology, Madras Lecture  36 Semiconductors We will start a discussion of semiconductors one of the most important
More informationMath 54. Selected Solutions for Week Is u in the plane in R 3 spanned by the columns
Math 5. Selected Solutions for Week 2 Section. (Page 2). Let u = and A = 5 2 6. Is u in the plane in R spanned by the columns of A? (See the figure omitted].) Why or why not? First of all, the plane in
More informationNotes: Most of the material presented in this chapter is taken from Bunker and Jensen (2005), Chap. 3, and Atkins and Friedman, Chap. 5.
Chapter 5. Geometrical ymmetry Notes: Most of the material presented in this chapter is taken from Bunker and Jensen (005), Chap., and Atkins and Friedman, Chap. 5. 5.1 ymmetry Operations We have already
More informationEach object can be assigned to a point group according to its symmetry elements:
3. Symmetry and Group Theory READING: Chapter 6 3.. Point groups Each object can be assigned to a point group according to its symmetry elements: Source: Shriver & Atkins, Inorganic Chemistry, 3 rd Edition.
More informationOrthogonal Projections
Orthogonal Projections and Reflections (with exercises) by D. Klain Version.. Corrections and comments are welcome! Orthogonal Projections Let X,..., X k be a family of linearly independent (column) vectors
More informationELEMENTS OF VECTOR ALGEBRA
ELEMENTS OF VECTOR ALGEBRA A.1. VECTORS AND SCALAR QUANTITIES We have now proposed sets of basic dimensions and secondary dimensions to describe certain aspects of nature, but more than just dimensions
More informationFinal Exam, Chem 311, 120 minutes, Dr. H. Guo, Dec. 17, You are allowed to bring a two page sheet containing equations and a calculator
Final Exam, Chem 311, 120 minutes, Dr. H. Guo, Dec. 17, 2008 You are allowed to bring a two page sheet containing equations and a calculator I. Answer the following multiple choice questions (5 pts each),
More informationSemiconductors, Insulators and Metals
CHAPTER 2 ENERGY BANDS AND EFFECTIVE MASS Semiconductors, insulators and metals Semiconductors Insulators Metals The concept of effective mass Prof. Dr. Beşire GÖNÜL Semiconductors, Insulators and Metals
More informationSimilarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
More informationLMB Crystallography Course, 2013. Crystals, Symmetry and Space Groups Andrew Leslie
LMB Crystallography Course, 2013 Crystals, Symmetry and Space Groups Andrew Leslie Many of the slides were kindly provided by Erhard Hohenester (Imperial College), several other illustrations are from
More informationSection 5.3. Section 5.3. u m ] l jj. = l jj u j + + l mj u m. v j = [ u 1 u j. l mj
Section 5. l j v j = [ u u j u m ] l jj = l jj u j + + l mj u m. l mj Section 5. 5.. Not orthogonal, the column vectors fail to be perpendicular to each other. 5..2 his matrix is orthogonal. Check that
More informationThe Quantum Harmonic Oscillator Stephen Webb
The Quantum Harmonic Oscillator Stephen Webb The Importance of the Harmonic Oscillator The quantum harmonic oscillator holds a unique importance in quantum mechanics, as it is both one of the few problems
More informationDiagonal, Symmetric and Triangular Matrices
Contents 1 Diagonal, Symmetric Triangular Matrices 2 Diagonal Matrices 2.1 Products, Powers Inverses of Diagonal Matrices 2.1.1 Theorem (Powers of Matrices) 2.2 Multiplying Matrices on the Left Right by
More information4.2. Linear Combinations and Linear Independence that a subspace contains the vectors
4.2. Linear Combinations and Linear Independence If we know that a subspace contains the vectors v 1 = 2 3 and v 2 = 1 1, it must contain other 1 2 vectors as well. For instance, the subspace also contains
More informationChapter 6. Orthogonality
6.3 Orthogonal Matrices 1 Chapter 6. Orthogonality 6.3 Orthogonal Matrices Definition 6.4. An n n matrix A is orthogonal if A T A = I. Note. We will see that the columns of an orthogonal matrix must be
More informationFree Electron Fermi Gas (Kittel Ch. 6)
Free Electron Fermi Gas (Kittel Ch. 6) Role of Electrons in Solids Electrons are responsible for binding of crystals  they are the glue that hold the nuclei together Types of binding (see next slide)
More informationElasticity Theory Basics
G22.3033002: Topics in Computer Graphics: Lecture #7 Geometric Modeling New York University Elasticity Theory Basics Lecture #7: 20 October 2003 Lecturer: Denis Zorin Scribe: Adrian Secord, Yotam Gingold
More informationThe Theory of Birefringence
Birefringence spectroscopy is the optical technique of measuring orientation in an optically anisotropic sample by measuring the retardation of polarized light passing through the sample. Birefringence
More informationAdvanced Techniques for Mobile Robotics Compact Course on Linear Algebra. Wolfram Burgard, Cyrill Stachniss, Kai Arras, Maren Bennewitz
Advanced Techniques for Mobile Robotics Compact Course on Linear Algebra Wolfram Burgard, Cyrill Stachniss, Kai Arras, Maren Bennewitz Vectors Arrays of numbers Vectors represent a point in a n dimensional
More informationAnother new approach to the small Ree groups
Another new approach to the small Ree groups Robert A. Wilson Abstract. A new elementary construction of the small Ree groups is described. Mathematics Subject Classification (2000). 20D06. Keywords. Groups
More informationMATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued).
MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors Jordan canonical form (continued) Jordan canonical form A Jordan block is a square matrix of the form λ 1 0 0 0 0 λ 1 0 0 0 0 λ 0 0 J = 0
More information1 2 3 1 1 2 x = + x 2 + x 4 1 0 1
(d) If the vector b is the sum of the four columns of A, write down the complete solution to Ax = b. 1 2 3 1 1 2 x = + x 2 + x 4 1 0 0 1 0 1 2. (11 points) This problem finds the curve y = C + D 2 t which
More informationWe consider a hydrogen atom in the ground state in the uniform electric field
Lecture 13 Page 1 Lectures 1314 Hydrogen atom in electric field. Quadratic Stark effect. Atomic polarizability. Emission and Absorption of Electromagnetic Radiation by Atoms Transition probabilities and
More informationMath 215 HW #6 Solutions
Math 5 HW #6 Solutions Problem 34 Show that x y is orthogonal to x + y if and only if x = y Proof First, suppose x y is orthogonal to x + y Then since x, y = y, x In other words, = x y, x + y = (x y) T
More informationSolutions to Linear Algebra Practice Problems
Solutions to Linear Algebra Practice Problems. Find all solutions to the following systems of linear equations. (a) x x + x 5 x x x + x + x 5 (b) x + x + x x + x + x x + x + 8x Answer: (a) We create the
More information4. MATRICES Matrices
4. MATRICES 170 4. Matrices 4.1. Definitions. Definition 4.1.1. A matrix is a rectangular array of numbers. A matrix with m rows and n columns is said to have dimension m n and may be represented as follows:
More information12.5 Equations of Lines and Planes
Instructor: Longfei Li Math 43 Lecture Notes.5 Equations of Lines and Planes What do we need to determine a line? D: a point on the line: P 0 (x 0, y 0 ) direction (slope): k 3D: a point on the line: P
More informationHow is a vector rotated?
How is a vector rotated? V. Balakrishnan Department of Physics, Indian Institute of Technology, Madras 600 036 Appeared in Resonance, Vol. 4, No. 10, pp. 6168 (1999) Introduction In an earlier series
More informationElectrical Conductivity
Advanced Materials Science  Lab Intermediate Physics University of Ulm Solid State Physics Department Electrical Conductivity Translated by MichaelStefan Rill January 20, 2003 CONTENTS 1 Contents 1 Introduction
More informationα = u v. In other words, Orthogonal Projection
Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v
More informationOrthogonal Matrices. u v = u v cos(θ) T (u) + T (v) = T (u + v). It s even easier to. If u and v are nonzero vectors then
Part 2. 1 Part 2. Orthogonal Matrices If u and v are nonzero vectors then u v = u v cos(θ) is 0 if and only if cos(θ) = 0, i.e., θ = 90. Hence, we say that two vectors u and v are perpendicular or orthogonal
More informationRepresentations of Molecular Properties
Representations of Molecular Properties We will routinely use sets of vectors located on each atom of a molecule to represent certain molecular properties of interest (e.g., orbitals, vibrations). The
More informationGroup Theory and Chemistry
Group Theory and Chemistry Outline: Raman and infrared spectroscopy Symmetry operations Point Groups and Schoenflies symbols Function space and matrix representation Reducible and irreducible representation
More informationx 3y 2z = 6 1.2) 2x 4y 3z = 8 3x + 6y + 8z = 5 x + 3y 2z + 5t = 4 1.5) 2x + 8y z + 9t = 9 3x + 5y 12z + 17t = 7 Linear AlgebraLab 2
Linear AlgebraLab 1 1) Use Gaussian elimination to solve the following systems x 1 + x 2 2x 3 + 4x 4 = 5 1.1) 2x 1 + 2x 2 3x 3 + x 4 = 3 3x 1 + 3x 2 4x 3 2x 4 = 1 x + y + 2z = 4 1.4) 2x + 3y + 6z = 10
More informationCrystal Structure Determination I
Crystal Structure Determination I Dr. Falak Sher Pakistan Institute of Engineering and Applied Sciences National Workshop on Crystal Structure Determination using Powder XRD, organized by the Khwarzimic
More informationThe University of Western Ontario Department of Physics and Astronomy P2800 Fall 2008
P2800 Fall 2008 Questions (Total  20 points): 1. Of the noble gases Ne, Ar, Kr and Xe, which should be the most chemically reactive and why? (0.5 point) Xenon should be most reactive since its outermost
More informationLECTURE SUMMARY September 30th 2009
LECTURE SUMMARY September 30 th 2009 Key Lecture Topics Crystal Structures in Relation to Slip Systems Resolved Shear Stress Using a Stereographic Projection to Determine the Active Slip System Slip Planes
More informationSolid State Theory Physics 545
Solid State Theory Physics 545 CRYSTAL STRUCTURES Describing periodic structures Terminology Basic Structures Symmetry Operations Ionic crystals often have a definite habit which gives rise to particular
More informationLecture 3: Optical Properties of Bulk and Nano. 5 nm
Lecture 3: Optical Properties of Bulk and Nano 5 nm First H/W#1 is due Sept. 10 Course Info The Previous Lecture Origin frequency dependence of χ in real materials Lorentz model (harmonic oscillator model)
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More informationDeriving character tables: Where do all the numbers come from?
3.4. Characters and Character Tables 3.4.1. Deriving character tables: Where do all the numbers come from? A general and rigorous method for deriving character tables is based on five theorems which in
More information0.1 Linear Transformations
.1 Linear Transformations A function is a rule that assigns a value from a set B for each element in a set A. Notation: f : A B If the value b B is assigned to value a A, then write f(a) = b, b is called
More informationApplied Linear Algebra I Review page 1
Applied Linear Algebra Review 1 I. Determinants A. Definition of a determinant 1. Using sum a. Permutations i. Sign of a permutation ii. Cycle 2. Uniqueness of the determinant function in terms of properties
More information